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I know that one way to do this is to evaluate the integral in the middle, and then compare these three numbers. I wonder how can we do this without explicitly compute the integral?

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Where can the e in the bound possible come from i wonder... – Lost1 Jan 4 '14 at 1:20
Integration by parts would seem to be one of the many magic bullets... – Igor Rivin Jan 4 '14 at 1:27
up vote 7 down vote accepted

Proof for the upper bound: $$\int_0^{\infty}e^{-x^2}dx =\int_{0}^{1}e^{-x^2}dx+\int_{1}^{\infty} e^{-x^2} dx <\int_{0}^{1}1 dx+\int_{1}^{\infty} x e^{-x^2} dx =1+\frac{1}{2e}$$ where the last integral is easily calculated by substituting $u=-x^2$.

Proof for the lower bound: $$\int_0^{\infty}e^{-x^2}dx > \int_{0}^{1}e^{-x^2}dx > \int_{0}^{1} xe^{-x^2} dx =\frac{1}{2}-\frac{1}{2e}$$ where the last integral is easily calculated by substituting $u=-x^2$.

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Oh nice. This is actually quite straight forward... – Lost1 Jan 4 '14 at 1:30
This is pretty awesome! Thanks!Really like this proof – Steven Jan 4 '14 at 1:42

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